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Yannick Privat; Emmanuel Trélat; Enrique Zuazua
On the best observation of wave and Schrödinger equations in quantum ergodic billiards
Journées équations aux dérivées partielles (2012), Exp. No. 10, 13 p., doi: 10.5802/jedp.93
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Class. Math.: 49J30, 49K20, 35L05, 93B05, 93C20
Mots clés: Wave equation, Schrödinger equation, observability inequality, optimal design, ergodic properties, Quantum Unique Ergodicity.

Résumé - Abstract

This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012.

In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set $\Omega \subset \mathbb{R}^n$, with Dirichlet boundary conditions. The observation is done on a subset $\omega $ of Lebesgue measure $\vert \omega \vert =L\vert \Omega \vert $, where $L\in (0,1)$ is fixed. We denote by $\mathcal{U}_L$ the class of all possible such subsets. Let $T>0$. We consider first the benchmark problem of maximizing the observability energy $\int _0^T\int _\omega \vert y(t,x)^2\, dx\, dt$ over $\mathcal{U}_L$, for fixed initial data. There exists at least one optimal set and we provide some results on its regularity properties. In view of practical issues, it is far more interesting to consider then the problem of maximizing the observability constant. But this problem is difficult and we propose a slightly different approach which is actually more relevant for applications. We define the notion of a randomized observability constant, where this constant is defined as an averaged over all possible randomized initial data. This constant appears as a spectral functional which is an eigenfunction concentration criterion. It can be also interpreted as a time asymptotic observability constant. This maximization problem happens to be intimately related with the ergodicity properties of the domain $\Omega $. We are able to compute the optimal value under strong ergodicity properties on $\Omega $ (namely, Quantum Unique Ergodicity). We then provide comments on ergodicity issues, on the existence of an optimal set, and on spectral approximations.


[1] N. Anantharaman, Entropy and the localization of eigenfunctions, Ann. of Math. 2 (2008), Vol. 168, 435–475.  MR 2434883 |  Zbl 1175.35036
[2] C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), no. 5, 1024–1065.  MR 1178650 |  Zbl 0786.93009
[3] F. Bonechi, S. De Bièvre, Controlling strong scarring for quantized ergodic toral automorphisms, Duke Math. J. 117 (2003), no. 3, 571–587.  MR 1979054 |  Zbl 1049.81028
[4] N. Burq, Large-time dynamics for the one-dimensional Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A. 141 (2011), no. 2, 227–251.  MR 2786680 |  Zbl 1226.35072
[5] N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 (2008), no. 3, 449–475.  MR 2425133 |  Zbl 1156.35062
[6] G. Chen, S.A. Fulling, F.J. Narcowich, S. Sun, Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math. 51 (1991), no. 1, 266–301.  MR 1089141 |  Zbl 0734.35009
[7] F. Faure, S. Nonnenmacher, S. De Bièvre, Scarred eigenstates for quantum cat maps of minimal periods, Comm. Math. Phys. 239 (2003), no. 3, 449–492.  MR 2000926 |  Zbl 1033.81024
[8] P. Gérard, E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J. 71 (1993), 559–607.  MR 1233448 |  Zbl 0788.35103
[9] A. Hassell, Ergodic billiards that are not quantum unique ergodic, Ann. Math. (2) 171 (2010), 605–619.  MR 2630052 |  Zbl 1196.58014
[10] A. Hassell, S. Zelditch, Quantum ergodicity of boundary values of eigenfunctions, Comm. Math. Phys. 248 (2004), no. 1, 119–168.  MR 2104608 |  Zbl 1054.58022
[11] P. Hébrard, A. Henrot, Optimal shape and position of the actuators for the stabilization of a string, Syst. Cont. Letters 48 (2003), 199–209.  MR 2020637 |  Zbl 1134.93399
[12] P. Hébrard, A. Henrot, A spillover phenomenon in the optimal location of actuators, SIAM J. Control Optim. 44 2005, 349–366.  MR 2177160 |  Zbl 1083.49002
[13] S. Kerckhoff, H. Masur, J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2) 124 (1986), no. 2, 293–311.  MR 855297 |  Zbl 0637.58010
[14] S. Kumar, J.H. Seinfeld, Optimal location of measurements for distributed parameter estimation, IEEE Trans. Autom. Contr. 23Ê(1978), 690–698.  Zbl 0381.93047
[15] V.F. Lazutkin, On the asymptotics of the eigenfunctions of the Laplacian, Soviet Math. Dokl. 12 (1971), 1569–1572.  Zbl 0232.35075
[16] G. Lebeau, Contrôle de l’equation de Schrödinger, J. Math. Pures Appl. 71 (1992), 267–291.  MR 1172452 |  Zbl 0838.35013
[17] K. Morris, Linear-quadratic optimal actuator location, IEEE Trans. Automat. Control 56 (2011), no. 1, 113–124.  MR 2777204
[18] Y. Privat, E. Trélat, E. Zuazua, Optimal observability of the one-dimensional wave equation, Preprint Hal (2012).
[19] Y. Privat, E. Trélat, E. Zuazua, Optimal location of controllers for the one-dimensional wave equation, Preprint Hal (2012).  MR 3048589
[20] Y. Privat, E. Trélat, E. Zuazua, Optimal observability of wave and Schrödinger equations in ergodic domains, ongoing work (2012).
[21] Z. Rudnick, P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), no. 1, 195–213.  MR 1266075 |  Zbl 0836.58043
[22] M. van de Wal, B. Jager, A review of methods for input/output selection, Automatica 37 (2001), no. 4, 487–510.  MR 1832938 |  Zbl 0995.93002
[23] S. Zelditch, Note on quantum unique ergodicity, Proc. Amer. Math. Soc. 132 (2004), 1869–1872.  MR 2051153 |  Zbl 1055.58016
[24] S. Zelditch, M. Zworski, Ergodicity of eigenfunctions for ergodic billiards, Comm. Math. Phys. 175 (1996), no. 3, 673–682.  MR 1372814 |  Zbl 0840.58048
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